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Chapter 4: Problem 47

Rewrite each sentence using mathematical symbols. Do not solve the equations. The product of 4 and the sum of a number and 6 is twice the number.

Short Answer

Expert verified
The equation is \(4(x + 6) = 2x\).

Step by step solution

01

Identify Variables and Operations

First, define the variable in the sentence. Let the unknown number be represented by the variable \( x \). Identify the key operations: multiplication (product of 4 and a sum) and addition (sum of a number and 6).
02

Construct the Sum Expression

The sum of the unknown number \( x \) and 6 can be written as an expression: \( x + 6 \).
03

Construct the Product Expression

The product of 4 and the expression \( x + 6 \) is written as: \( 4(x + 6) \).
04

Translate 'Twice the Number'

The phrase "twice the number" means multiplying the number \( x \) by 2. This can be written as: \( 2x \).
05

Write the Equation

Combine these expressions into an equation. The product of 4 and the sum of a number and 6 (\(4(x + 6)\)) equals twice the number (\(2x\)). The equation is: \[ 4(x + 6) = 2x \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Variables in Algebra
In algebra, variables are symbols that represent unknown values in a mathematical expression or equation. These symbols are generally letters of the alphabet, like \( x \), \( y \), or \( z \). They serve as placeholders for numbers that we are yet to determine.
  • Defining Variables: In the given exercise, the letter \( x \) is used to stand for 'a number'. We don't know what this number is, which is why we use a variable to represent it.
  • Why Use Variables: Using variables can help us model real-world situations where exact numbers are variable or unknown, enabling us to solve a wide range of problems.
Variables are essential for forming expressions, constructing equations, and finding solutions in algebra.
Constructing Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operations (adds, subtracts, multiplies, divides). Constructing expressions involves representing a word problem or situation mathematically.
  • Addition Expressions: In the example exercise, 'the sum of a number and 6' is translated to \( x + 6 \).
  • Multiplication Expressions: 'The product of 4 and the sum' translates to 4 times the expression, which is written as \( 4(x + 6) \).
Constructing expressions is vital as it transforms a verbal problem statement into a mathematical form, making it easier to analyze and solve.
Equations in Algebra
An algebraic equation is like a balance, depicting two equal expressions separated by an equals sign \( = \). It tells us that two mathematical expressions represent the same quantity.
  • Understanding Equality: In the problem, the equation states that the product of 4 and the sum of a number and 6 is equal to twice the number, represented by \( 4(x + 6) = 2x \).
  • Components of an Equation: On the left, we have \( 4(x + 6) \), which means we multiply 4 by \( x + 6 \). On the right, \( 2x \) implies doubling \( x \).
Equations help us work out what a variable stands for by setting different expressions equal to each other, which we then solve through algebraic manipulations.

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Most popular questions from this chapter

Use a graphing calculator to solve each system. $$ \left\\{\begin{array}{l} y=3.1 x-16.35 \\ y=-9.7 x+28.45 \end{array}\right. $$

Solve each system of equations by the addition method. If a system contains fractions or decimals, you may want to first clear each equation of fractions or decimals. $$ \left\\{\begin{array}{l} -2.5 x-6.5 y=47 \\ 0.5 x-4.5 y=37 \end{array}\right. $$

Solve each system of equations by the addition method. If a system contains fractions or decimals, you may want to first clear each equation of fractions or decimals. $$ \left\\{\begin{array}{l} 6 x-5 y=25 \\ 4 x+15 y=13 \end{array}\right. $$

Solve each system of equations by the addition method. If a system contains fractions or decimals, you may want to first clear each equation of fractions or decimals.$$ \left\\{\begin{array}{l} x-y=1 \\ -x+2 y=0 \end{array}\right. $$

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} 3 x+y=-14 \\ 4 x+3 y=-22 \end{array}\right. $$
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